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Constantinou (2006) has observed that the tensile stiffness of the elastomer remains the same as in the case of compressive loading until 20.6-MPa stress. A rapid development in the tensile strain beyond the 20.6-MPa stress has been observed. In general, tensile strain has been found to be in the range of 200–600% at the time of failure of the elastomer (Constantinou 2006). In this paper, the tensile stiffness of the elastomer has been assumed to be the same as the compressive stiffness, and the tensile strength has been used to control the failure of bearings to match the observed behavior during experiments. For shear failure of the bearings, 150% shear strain in the elastomer is used as the failure criterion. Parameters of the elastomer used in the finite-element model of the bearing include: bulk modulus 5 82.73 MPa; shear modulus 5 593 kPa; tensile failure strain 5 30.5%; and compressive failure stress 5 140 MPa. The plastic-kinematic material model (material Mat 3 in LS-DYNA) has been used for steel shims, modeling isotropic and kinematic hardening plasticity with the option of high-strain rate effects.
For the elastomeric model of the bearing in Fig. 13(c), Table 5 shows a comparison between the properties of elastomeric bearings observed through numerical simulation to those from experimental results (Muscarella and Yura 1995). It is observed that the compressive modulus and tensile strength predicted by the FEM model match well with experimental results reported by Muscarella and Yura (1995). The computational compressive strength of the bearing falls on the lower side of the range of variation of 12,300–20,300 psi observed during the experiments. The compressive strength of bearings during experiments varies because of the yielding of steel shims and hardening of the steelmaterial. Because the elastoplastic stress-strain relationship is assumed for steel material, the bearing fails quickly due to the bulging of the elastomer after initial yielding of the steel shims. Therefore, a lower compressive strength in the numerical simulation is reasonable. On the other hand, bearings during numerical simulations fail at 35.6% tensile strain, whereas tensile failure strain of the elastomer during experiments has been observed in the range of 200–600% even though tensile failure strength is the same. Likewise, shear failure strain for the bearings during numerical simulations is 61.9%, whereas shear strain failure criteria for the elastomer has been considered to be 150%. The bearing fails at lower values of tensile and shear strains because of delamination between the elastomers and steel shims before complete tensile or shear failure of the elastomeric materials. Hence, the FEM model of bearings in Fig. 13(c) will provide a reasonably accurate force in the simulation of bridge performance.
Table 5. Validation of FEM Bearing Model
Finite-Element Model of the Whole Bridge
A detailed finite-element model of the complete bridge has been developed in LS-DYNA. First of all, bridge footings are constructed using solid elements and fixed boundaries at the bottom of the footings, as shown in Fig. 16(a). A rebar cage for concrete piers has been developed by modeling rebars by beam elements, as shown in Fig. 16(b). Rebars extend into the footings and bent beams per rebar detailing on the as-built bridge drawings. Fig. 16(c) shows the rebar detailing in the footings, pier, and bent beam. Core concrete is added to the rebar cage, and cover concrete is added as a separate layer withdifferent material properties on top of the core concrete.Abent beam is added on top of the concrete piers.
Fig. 16. FEM modeling of bridge under blast load in LS-DYNA: (a) add footing; (b) add rebar cage for concrete pier; (c) add core concrete, surface concrete of concrete pier, and pier bent; (d) add bearing; (e) add stringer; (f) add deck; (g) put the bridge model into air ALE mesh
Elastomeric bearings aremodeled as a block consisting of layers of elastomers and steel shims connected by shared nodes, as described in the previous section. Elastomeric-bearing blocks are added on top of pier bents at locations specified in bridge drawings.Astiff elastomeric bearing, which behaves like a fixed bearing, is installed at the south pier away from the explosion. Regular elastomeric bearings, which act as expansion bearings, are installed at the other three piers and abutments. Bearing blocks are connected to the piers through shared nodes, which model the connection similar to anchor bolts. Steel stringers and diaphragms,modeled by shell elements, are added next, as shown in Fig. 16(e). Steel stringers are connected to elastomericbearing blocks through shared nodes. A deck, constructed by using solid elements is added next, as shown in Fig. 16(f).